DS on What is a Line Axiomathes

We start from the history of " continuum problem " and study of infinitely small and infinitely large numbers, rediscover the semi-ring of tropical real numbers and also reconfirm the philosophy that functions are generalized numbers. Alongside, we discuss the " number-shape problem " stating that, numbers and geometric shapes have the same essence.

The aim of this paper is to understand Aristotle's idea of continuity as unity and infinity. The philosopher uses the idea of continuity in two ways: In Physics V (3) he employs the term " continuity " in the sense of holistic unity. At 227a10 he writes, " something is continuous [συνεχές] when two things touch each other by their limits and become one and the same thing and, as the name indicates, hold themselves together [συνέχηται]. " In Physics VI (1), however, he employs the term " continuous " in the sense of infinite divisibility. At 231a21-29 he writes, " it is impossible for a continuum to consist of indivisible things. For example, a continuous line, which is infinitely divisible, cannot consist of points, which are indivisible and discontinuous. " Here, Aristotle is using " continuous " to refer to the property that belongs to a single

The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. This article explains the distinction and why it has proved to be one of the great organizing themes of mathematics.

I consider some ways in which our intuitive understanding of nature of line and linearity have exerted an influence on how we conceive of reality in general. My principle topic is something of fundamental importance: the nature of space and time.

“Infinitesimal - A Dangerous Mathematical Theory” by Amir Alexander (Scientific American/Farrar – 2014) © A review essay by H. J. Spencer May 2017 This book is about one of the most dangerous ideas ever invented by human beings. It led to the theory that eventually destroyed the monopoly hold on the educated minds of Western Europe by planting one of the principal seeds of modern physics. Award-winning UCLA historian, Alexander has written a fascinating tale of how abstract ideas influence society. This book deserves to be read not just by scientists and engineers, whose lives are dominated by mathematics, but by any educated person who had to suffer through calculus class. This book is about one of the most dangerous ideas ever invented by human beings. It led to the theory that eventually destroyed the monopoly hold on the educated minds of Western Europe by planting one of the principal seeds of modern physics. Award-winning UCLA historian, Amir Alexander has written a fascinating tale of how abstract ideas influence society. This book deserves to be read not just by scientists and engineers, whose lives are dominated by mathematics, but by any educated person who had to suffer through calculus class. This book is much more than an esoteric history of an area of mathematics. It tracks the ancient rivalry between 'rationalists' and 'empiricists'. The dominant rationalists have always believed that human minds (at least those possessed by educated intellectuals) are capable of understanding the world purely by thought alone. The empiricists acknowledge that reality is far too complicated for humans to just guess its detailed structures. This is not simply an esoteric philosophical distinction but the difference in fundamental world-views that have deeply influenced the evolution of western civilization. In fact, rationalist intellectuals have usually looked to the logical perfection of mathematics as a justification for the preservation of religion and hierarchical social structures. In particular, the rationalists have raised the timeless, unchanging mathematical knowledge, represented by Euclidean geometry, as not just the only valid form of symbolic knowledge but as the only valid model of the logic of " proof ". In particular, this book focuses on the battle between the reactionaries (e.g. Jesuits and Hobbes), who needed a model of timeless perfection to preserve their class-based religious and social privileges and reality-driven modernists, like Galileo and Bacon, who were desirous of major changes. In the late Middle-Ages, the new order of Jesuits were the intellectual leaders of the Catholic Church and were formed to defeat the recent Reformation. They not only opposed Protestant theology but also the parallel forces of pluralism, populism and social reform. The Jesuits, like their Church itself and the ancient social structures they supported, were all organized on traditional (militaristic) hierarchical principles. In 1632, the Jesuits convened a major council in Rome and decided to ban the idea of " indivisibles " – the old idea that a line was composed of distinct and an infinity of tiny parts. They correctly anticipated that the threat of this idea to their rational view of the world, as an " orderly place, governed by a strict and unchanging set of rules. " Geometry was their best exemplar of their Catholic theology. The core of the disagreement was over the nature of the continuum, a concept that had surfaced in Ancient Greece. The reality of the idea of 'physical indivisibles' (atoms) was even being disputed by serious scientists as late as 1900. The original idea of continuity was stimulated by the apparent lack of observable 'gaps' in solids or liquid materials and our personal sense of the continuous flow of time (ideally, also endless). This idea became a key concept to many Ancient Greek thinkers; even, Aristotle made the plausible statement that: " No continuous thing is divisible into parts. " This was soon 'cast in concrete' with Euclid's major definition of a line as an infinite number of points. This became one of the core (obvious) assumptions of geometry – the basis of so much of western education. The concept of 'Continuity' became a key Principle of medieval scholastic thought, eagerly latched onto by Aquinas and other theologians in their battle with the atheistic and equally ancient idea of atoms.

The continuity and the discreetness in philosophy ... the gap between the continuous and the distinct "this permanent obstacle" that plays an important role in mathematics, philosophy and physics (A Fraenkel (Vilenkin 1997)c We are all familiar with the idea of continuity. To be continuous [ is to constitute an unbroken or uninterrupted whole, like the ocean or the sky. A continuous entity-a continuum-has no-gaps‖. Opposed to

Technical Transactions. Kraków. – 2014. – 1-NP. – p. 195-209., 2014

The notion of connectivity was introduced by Listing in 1847; it was developed by Riemann, Jordan, Poincaré. The notion and rigorous definition of metric and topological space was formed in Frechet’s works in 1906, and Hausdorff’s works in 1914. The notion of continuum could be traced back to Antiquity, but its mathematical definition was formed in XIX century, in the works of Cantor, Dedekind, later Hausdorff and Riesz. Karl Weierstrass (1815-1897) brought mathematical analysis to a rigorous form, whereas the notions of future mathematics – functional analysis and topology were formed in his reasoning. Weierstrass’s works were not translated into Russian, and his lectures were not published even in Germany. In 1989, synopses of his lectures devoted to additional chapters of the theory of functions were published, which material has served as the basis for this article.

Real Numbers, Generalizations of the Reals, and Theories of Continua, 1994

2002

Computations in geographic space are necessarily based on discrete versions of space, but much of the existing work on the foundations of GIS assumes a continuous infinitely divisible space. This is true both of quantitative approaches, using ℝ n, and qualitative approaches using systems such as the Region-Connection Calculus (RCC). This paper shows how the RCC can be modified so as to permit discrete spaces by weakening Stell's formulation of RCC as Boolean connection algebra to what we now call a connection algebra.